Cassirer and Klein on the Geometrical Foundations of Relativistic Physics

Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from the spacetimes of classical mechanics and of special relativity to those of general relativity. Not only is it much more complicated to find such invariants in the case of general relativity, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of general relativity seems to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. This paper aims to reconsider these issues by drawing attention to the combination of subtractive and additive strategies (more in line with Riemann’s) in Klein’s own considerations about the determination of the structure of spacetime from 1897 to 1910. I will point out that Cassirer relied on Klein’s argument in some central passages from Substance and Function (1910), and elaborated further on his combined approach in Einstein’s Theory of Relativity (1921), also taking into account the application of Riemannian geometry in general relativity. My suggestion is that an appreciation of Cassirer’s continuing commitment to a variety of geometrical traditions may shed light on his particular understanding of a priori elements of knowledge and avoid some of the classical objections to the idea of a relativization of the a priori.